The Game Consists of Elements
The author defines four basic elements of a game: mechanics, story, aesthetics, and technology. He stresses that all of the elements are important, and each influences the others, but I really think that's pushing it.
However, he notes that each of them should be considered from the perspective of audience, and what they consider to be important. A game without a story and that has poor graphics may be acceptable to some, but not others.
Mechanics
Mechanics are the rules and procedures of the game that describe the goal , what actions the player may take to achieve it, and what actions the opponents may take to make achievement of the goal more difficult.
The mechanics are generally what makes the game compelling, and achieving the right balance can be difficult.
Story
The sequence of events that unfold in the game, including the scripted events and those that are emergent (appear/evolve within the context of the game).
A game doesn't need to have a story (irony: the author said all four elements are important, but now says this one is optional). However, the "story" gives the game its identity and is generally the way in which a game is described to those who haven't played it.
Note that games with similar mechanics are often repackaged with a different story: whether the player's "piece" is a submarine, airplane, or space ship, the mechanics of the game is similar.
Aesthetics
The look-and-feel of the physical artifacts (or digital artifacts) of the game. In the videogame space, this includes visual and audio elements.
Aesthetics can be dismissed as style (recall how compelling the games designed for the original Atari system were, even though their appearance is primitive by modern standards), but at the same time, they give the user a sense of the quality of the game.
Technology
Not computer technology (programming languages, etc.), but the physical artifacts of the game. This may be pencil and paper or a board and pieces. In the video gaming space, it's generally the computer and controller.
In video gaming, much of this is determined by the terminal (computer, game console) design, unless the game design also includes hardware (and the hardware can have novelty value).